PERIODIC MOTION 299 But since sin 0 is a periodic function of 6 with a period of 2 T, we have sin 6= sin (<9+27r), therefore x = a sin [w (J + P) + 5 ] = a sin [&rf+5+2ir] and consequently (IV) 231. Frequency. — The number of complete vibrations which the particle makes per second is called the frequency of the vibration. If n denotes the frequency, then n-i-£. (V) 232. Time-distance Diagram. — Suppose the particle to describe the vertical line A A', Fig. 131, the middle point of FIG. 131. which is the fixed point. Then OA = OA! = a = the amplitude. The relation between the position of the particle and the time may be visualized by plotting equation (III) with x as ordinate and t as abscissa. This gives the well-known sine curve. A mental picture of the motion of "the particle may be formed by supposing that the particle under consideration is a projection of another particle which moves in a circle of radius a with a constant speed. The second particle and its path may be called the auxiliary particle and the auxiliary j respectively.